>>854

(3/4)(1 + A/H)^2 - (A/G)^3 - (G/H)^3 -1 = (1/108){(a-b)(b-c)(c-a)/abc}^2 ≧ 0,

(略証)
s = a+b+c,　t = ab+bc+ca,　u = abc,　⊿ = (a-b)(b-c)(c-a),
とおくと
A = s/3,　G = u^(1/3),　H = 3u/t,
A/H = st/9u,　A/G = s/{3u^(1/3)},　G/H = t/{3u^(2/3)},
ゆえ
(左辺) = (3/4)(1+st/9u)^2 - s^3/27u - t^3/27uu -1
　= (1/108uu){(st+9u)^2 -4s^3u -4t^3 -108uu}
　= (1/108uu)⊿^2,